Finding the Average Rates of Change in Repetitive Behavior

The repetitive behavior of a device or system can be described in two ways: a detailed description of one iteration of the behavior, or a summary description of the behavior over many repetitions. This paper describes an implemented program called AIS that transforms the first type of description into the second type. AIS deals only with behavior where each repetition changes parameters by the same amounts. At present, the summary consists of the symbolic average rates of change in parameter values and information on how those rates would be different if various constants and functions had been different. Unlike some other approaches, AIS does not require that a repeating behavior be described in terms of a set of differential equations. Two examples of running AIS are given: one concerns the human heart, the other a steam engine. INTRODUCTION: I have implemented a program called AIS (short for Analyzer of Iterated Sequences) that when given a continuous state-description of a system and a sequence of actions or transformations on that state, symbolically finds some of the time-averaged effects of continually iterating that sequence. The specific effects found at present include 1) the symbolic average rate of change in parameters that have a net increase or decrease in value with each iteration, and 2) how those rates of change would be different with different values of various constants and functions (sensitivity analysis). The sequences handled by AIS are ones which have the following “constancy”: the sequence always has the same actions in the same order and each occurrence of a particular action changes the parameters by the same amounts. An example of such an ∗Supported by the National Institute of Health through grant R01-LM04493 from the National Library of Medicine and grant R01-HL33041 from the National Heart, Lung, and Blood Institute. iterated action sequence is the one taken by a heart in going through a beat cycle at steady-state. Effects to be found include the average rate at which blood enters the heart and how increasing the pressure of that entering blood affects that rate. A motivation for finding such effects is that while modeling some system, there may be some sub-system β which iterates a sequence of actions at such a fast rate that the rest of the system only responds to β’s behavior averaged over many iterations. Then a steadystate model for the entire system would only need a description of β’s averaged behavior; β can be modeled as constantly iterating the same sequence of parameter value changes. Examples of such sub-system and system combinations include 1) the heart and the human circulatory system, and 2) an engine and a car. Some other approaches of finding the behaviors of a continually iterating sequence have combined qualitative simulation with cycle detection [1]. For complicated systems (such as the heart), these simulations predict many possible sequences of actions besides the actual sequence. If the actual sequence can be isolated, one can use aggregation [10] to find which parameters change as the sequence repeats and use comparative analysis [11] to find the effects of perturbing model constants. Another approach [6] uses piecewise-linear approximations of differential equations. This approach requires that one describe a system with a single set of always applicable differential equations. Creating such a description may often be hard, such as when describing a human heart or a steam engine. In contrast, the input for both the qualitative simulation approaches and AIS can have many sets of simple equations along with the conditions to determine when a particular set is applicable. The next section describes the form of input for AIS. Following this are sections on how AIS processes that input and on what AIS can output. Afterwards are sections that give examples of AIS running on a description of a heart and a steam engine, respectively. The paper ends with a summary.